CROSS-VALIDATION AND MODEL SELECTION Many Slides are from: Dr. Thomas Jensen -Expedia.com and Prof. Olga Veksler - CS Learning and Computer Vision

How to check if a model fit is good? The R 2 statistic has become the almost universally standard measure for model fit in linear models. What is R 2 ? It is the ratio of error in a model over the total variance in the dependent variable. Hence the lower the error, the higher the R2 value.

How to check if a model fit is good?

OVERFITTING Modeling techniques tend to overfit the data. Multiple regression: Every time you add a variable to the regression, the model’s R 2 goes up. Naïve interpretation: every additional predictive variable helps to explain yet more of the target’s variance. But that can’t be true! Left to its own devices, Multiple Regression will fit too many patterns. A reason why modeling requires subject-matter expertise.

OVERFITTING Error on the dataset used to fit the model can be misleading › Doesn’t predict future performance. Too much complexity can diminish model’s accuracy on future data. › Sometimes called the Bias- Variance Tradeoff.

OVERFITTING What are the consequences of overfitting? ›“Overfitted models will have high R 2 values, but will perform poorly in predicting out-of-sample cases”

WHY WE NEED CROSS-VALIDATION? R squared, also known as coefficient of determination, is a popular measure of quality of fit in regression. However, it does not offer any significant insights into how well our regression model can predict future values. When an MLR equation is to be used for prediction purposes it is useful to obtain empirical evidence as to its generalizability, or its capacity to make accurate predictions for new samples of data. This process is sometimes referred to as “validating” the regression equation.

One way to address this issue is to literally obtain a new sample of observations. That is, after the MLR equation is developed from the original sample, the investigator conducts a new study, replicating the original one as closely as possible, and uses the new data to assess the predictive validity of the MLR equation. This procedure is usually viewed as impractical because of the requirement to conduct a new study to obtain validation data, as well as the difficulty in truly replicating the original study. An alternative, more practical procedure is cross-validation.

CROSS-VALIDATION In cross-validation the original sample is split into two parts. One part is called the training (or derivation) sample, and the other part is called the validation (or validation + testing) sample. 1)What portion of the sample should be in each part? If sample size is very large, it is often best to split the sample in half. For smaller samples, it is more conventional to split the sample such that 2/3 of the observations are in the derivation sample and 1/3 are in the validation sample.

CROSS-VALIDATION 2) How should the sample be split? The most common approach is to divide the sample randomly, thus theoretically eliminating any systematic differences. One alternative is to define matched pairs of subjects in the original sample and to assign one member of each pair to the derivation sample and the other to the validation sample. Modeling of the data uses one part only. The model selected for this part is then used to predict the values in the other part of the data. A valid model should show good predictive accuracy. One thing that R-squared offers no protection against is overfitting. On the other hand, cross validation, by allowing us to have cases in our testing set that are different from the cases in our training set, inherently offers protection against overfitting.

CROSS VALIDATION – THE IDEAL PROCEDURE 1.Divide data into three sets, training, validation and test sets 2.Find the optimal model on the training set, and use the test set to check its predictive capability 3.See how well the model can predict the test set 4.The validation error gives an unbiased estimate of the predictive power of a model

TRAINING/TEST DATA SPLIT Talked about splitting data in training/test sets training data is used to fit parameters test data is used to assess how classifier generalizes to new data What if classifier has “non‐tunable” parameters? a parameter is “non‐tunable” if tuning (or training) it on the training data leads to overfitting

TRAINING/TEST DATA SPLIT What about test error? Seems appropriate degree 2 is the best model according to the test error Except what do we report as the test error now? Test error should be computed on data that was not used for training at all Here used “test” data for training, i.e. choosing model

VALIDATION DATA Same question when choosing among several classifiers our polynomial degree example can be looked at as choosing among 3 classifiers (degree 1, 2, or 3) Solution: split the labeled data into three parts

TRAINING/ VALIDATION

Training/Validation/Test Data Training Data Validation Data d = 2 is chosen Test Data 1.3 test error computed for d = 2

LOOCV (Leave ‐ one ‐ out Cross Validation) For k=1 to R 1. Let (x k,y k ) be the k example

LOOCV (Leave ‐ one ‐ out Cross Validation)

LOOCV for Quadratic Regression

LOOCV for Join The Dots

Which kind of Cross Validation?

K-FOLD CROSS VALIDATION ›Since data are often scarce, there might not be enough to set aside for a validation sample ›To work around this issue k-fold CV works as follows: 1. Split the sample into k subsets of equal size 2. For each fold estimate a model on all the subsets except one 3. Use the left out subset to test the model, by calculating a CV metric of choice 4. Average the CV metric across subsets to get the CV error ›This has the advantage of using all data for estimating the model, however finding a good value for k can be tricky

K-fold Cross Validation Example 1.Split the data into 5 samples 2.Fit a model to the training samples and use the test sample to calculate a CV metric. 3.Repeat the process for the next sample, until all samples have been used to either train or test the model

Which kind of Cross Validation?

Improve cross-validation Even better: repeated cross-validation Example: 10-fold cross-validation is repeated 10 times and results are averaged (reduce the variance)

Cross Validation - Metrics How do we determine if one model is predicting better than another model?

Cross Validation Metrics

Best Practice for Reporting Model Fit 1.Use Cross Validation to find the best model 2.Report the RMSE and MAPE statistics from the cross validation procedure 3.Report the R Squared from the model as you normally would. The added cross-validation information will allow one to evaluate not how much variance can be explained by the model, but also the predictive accuracy of the model. Good models should have a high predictive AND explanatory power!